This contract is part of a study of T cell-depletion in unrelated donor bone marrow transplantation. The primary objective is to determine if a reduction in morbidity and mortality from acute and chronic graft versus host disease (GvHD) can be achieved in patients given T cell-depleted marrow from an unrelated donor without a counterbalancing increase in relapse of leukemia and graft failure. This prospective, randomized study will require a total of 560 patients. Study investigators will randomize patients to T cell-depletion or no-depletion. Each therapy group will utilize their own method of T cell-depletion. Two important components of this study are to characterize the transplanted marrow with regard to numbers and types of cells in the graft, and to monitor immune reconstitution. This project includes 4 therapy groups and a medical coordinating center. This study is expected to last seven and one-half years.